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Math Help - Another contrived counting problem...

  1. #1
    Junior Member NoFace's Avatar
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    Another contrived counting problem...

    I just don't know how to set this one up. Here it is:

    Five married couple stand in a row so that no wife stands next to her own husband. In how many ways can they stand?

    Seems like I should throw a derangement in there. But I think using inclusion-exclusion would be easier.

    Thanks.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by NoFace View Post
    Five married couple stand in a row so that no wife stands next to her own husband. In how many ways can they stand?
    This is indeed an inclusion-exclusion problem.
    Couple I can stand together in (9!)(2) ways.
    Couples I&II can stand together in (8!)\left( {2^2 } \right) ways.
    Consider the number of ways that at least one couple is together and subtract from the total.
    \sum\limits_{k = 0}^5 {\left( { - 1} \right)^k {5 \choose k}\left[ {(10 - k)!\left( {2^k } \right)} \right]} .
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